| L(s) = 1 | + 2·25-s + 2·49-s + 4·73-s − 4·97-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 2·25-s + 2·49-s + 4·73-s − 4·97-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.338960901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.338960901\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$ | \( ( 1 - T )^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$ | \( ( 1 + T )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.256138654178166197316001342571, −9.030360332210759474978608301349, −8.618582661939724690539161761209, −8.306522801608302811399082525584, −7.77869817667419196365074396252, −7.59670940442531178970512800069, −6.88036566963366407484521607269, −6.69352018234265738754400787846, −6.53062603387413204559482472777, −5.74614482141289107057794243746, −5.35516703300717602532838583971, −5.20531561412361022727792090914, −4.51324427484928795713985504553, −4.22310695014203753438899659805, −3.64167956491301591203916541276, −3.25576330057765085835529047619, −2.49320609828736547236567061822, −2.42610052844155881412945048433, −1.43762214026917881073060627354, −0.871239430364633848829850410997,
0.871239430364633848829850410997, 1.43762214026917881073060627354, 2.42610052844155881412945048433, 2.49320609828736547236567061822, 3.25576330057765085835529047619, 3.64167956491301591203916541276, 4.22310695014203753438899659805, 4.51324427484928795713985504553, 5.20531561412361022727792090914, 5.35516703300717602532838583971, 5.74614482141289107057794243746, 6.53062603387413204559482472777, 6.69352018234265738754400787846, 6.88036566963366407484521607269, 7.59670940442531178970512800069, 7.77869817667419196365074396252, 8.306522801608302811399082525584, 8.618582661939724690539161761209, 9.030360332210759474978608301349, 9.256138654178166197316001342571
